Talk:Parseval's identity

Parseval's Equality

This equality is wrong, that's also why it does not work to prove the Basel problem. 86.95.192.7 (talk) 08:52, 8 November 2008 (UTC)[reply]

Are you sure? I've checked it myself, and it seems to work out. Make sure you're using the same definitions everwhere- you can put the multiplicative constants in different places. 158.59.247.77 (talk) 19:48, 3 April 2009 (UTC)[reply]

Incorrect formula

The coefficient of the integral in the identity should be 1/pi, not 1/(2*pi). —Preceding unsigned comment added by 98.233.155.147 (talk) 02:34, 6 April 2009 (UTC)[reply]

No. The formula is correct. What is perhaps confusing is that it is customary to define real Fourier coefficients

a_{n}

and

b_{n}

by a formula with

1/\pi

instead of the

1/(2\pi )

of the complex Fourier coefficients. Perhaps should one rename the complex coefficients here to be

c_{n}

, as it is quite usual. --Bdmy (talk) 11:55, 6 April 2009 (UTC)[reply]

I fixed the identity, it was wrongly written previously as $\|f\|_{2}^{2}=\int _{-\pi }^{\pi }|f(x)|^{2}\,dx=2\pi \sum _{n\in \mathbb {Z} }|{\hat {f}}(n)|^{2}$ However, in the context of the article, the Hilbert space is $L^{2}([-\pi ,\pi ),\mathbb {C} ))$ with inner product given by $\langle f,g\rangle :={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(x){\overline {g(x)}}\,dx$ so here it must be $\|f\|_{2}^{2}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }|f(x)|^{2}\,dx=\sum _{n\in \mathbb {Z} }|{\hat {f}}(n)|^{2}$

Also in this context it must be $e_{n}=e^{inx}$ , and not $e_{n}={\frac {e^{-inx}}{2\pi }}$ as it was previously written.

Edit war over "Hesham's identity"

Hesham Sarkas,

You and I appear to be beginning an edit war regarding whether the the appropriateness of your additions to this page.

(English) Wikipedia has a long-standing policy regarding the information in Wikipedia pages: specifically, Wikipedia pages should only summarize material contained in reliable and verifiable secondary or tertiary sources, reproducing the emphases of the mainstream scholarly literature. In particular, an uncited, self-published, non-peer-reviewed preprint is too unreliable a source to justify the inclusion of material in the encyclopedia.

Moreover, Wikipedia explicitly states that contributors should, in general, refrain from editing articles to which they have a personal connection, and use "material you have written or published...only if it is relevant, conforms to the content policies, including WP:SELFPUB, and is not excessive." I submit that the content you wish to add is excessive and irrelevant, and fails WP:SELFPUB (your publication record at this time is too spotty to consider you a subject-matter expert). I have never seen the term "Hesham's identity" before, despite owning multiple books treating Fourier analysis.

Moreover, the so-called "Hesham's identity" appears to be a routine computation, expanding a (position-space) multiplication as a Fourier-space convolution. Routine computations are acceptable on Wikipedia, but only when the computation is necessary to integrate source material into Wikipedia's existing treatment. Such is not the case here. Moreover, the routine calculation exception does not cover named results; a reliable source is required to indicate that the term "Hesham's identity" is in common use for this particular algebraic manipulation.

If you have questions, {{ping}} me and I'd be happy to answer them; if you disagree with my assessment of the content, feel free to get a second opinion with another editor (or ask me to find another editor for one).

Best of luck in your future endeavors, Bernanke's Crossbow (talk) 23:21, 4 January 2023 (UTC)[reply]

Hello Bernanke's Crossbow

Firstly, thanks for your time and thorough assessment.I can see you are a PhD student in pure mathematics, that is why you don't know why the added formula is useful. If you have worked before with artificial intelligence or signal processing you would know that the two formulas I added are very useful and very efficient with simulation rather than numerically calculating the integral. You said the preprint is a routine computation, and basicly any theorem or identity in mathematics has a routine calculation proof so no thing wrong about that. The proof I referenced that you call (A routine calculation) no one did it before, and you confirmed that you never seen the identity in any book, this makes it a new identity proofed with calculations. If you claim the proof is too "spotty" then why you didn't do it before? As you said you never seen the identity before in any book, and you didn't find any mathematical mistakes in it -otherwise you would say- then let Signal processing people see it and use it. The preprint is in the stage of consideration in a peer-reviewed journal, and soon will be published, and then I will add the published paper as a reference. Besides, you are not the first to contact me about the identity, because others did contact to report that they found it useful, so leave it public and don't keep removing it and soon I will replace the preprit with the published version. 102.47.70.163 (talk) 03:47, 5 January 2023 (UTC)[reply]

Hesham,

I see. I know making it through the publication process can be tricky; I hope it goes smoothly for you.

Bernanke's Crossbow (talk) 18:06, 7 January 2023 (UTC)[reply]

A lot of anonymous editors are keen to add this here and at Fourier series. From a policy perspective this is a non-starter: additions like this require secondary sources that are independent of the concepts originator. On a mathematical level, this is a trivial identity on trigonometric polynomials, not Fourier series. In particular, the $L^{4}$ requirement is a complete red herring: every trigonometric polynomial is in $L^{4}$ (in fact $L^{\infty }$ , or even every Sobolev space). Perhaps an even bigger problem with this though, is that the Fourier series in L^4, unlike the case of L^2, do not map into $\ell ^{4}$ , so this identity would not hold in L^4. Tito Omburo (talk) 11:41, 5 November 2023 (UTC)[reply]

Tito Omburo,

From a policy perspective, I will keep your version till the completion of the publish process then revert to previous version with this section included. Hopefully by the end of the year. From Mathematical level, the identity is already in use in signal processing, image processing and Machine learning. You seem to be a pure mathematics guy with low hands-on experience so you can't understand why. The problem you raised is interesting, and thanks for your feedback. 45.137.113.91 (talk) 14:40, 5 November 2023 (UTC)[reply]

You should get consensus here on the discussion page before restoring the content. Normally things like this require coverage in secondary sources. A single published source by the originator of the concept is not enough. If there are sources in evidence that this identity is "already in use in signal processing, image processing and Machine Learning", please list those sources for discussion. Not that my background is relevant,I am an applied mathematician specializing in probability and mathematical analysis. Tito Omburo (talk) 15:03, 5 November 2023 (UTC)[reply]

You didn't get consensus before you take away the contents although many editor reverted your action. Another articles in SP, IP, and ML utilize the identity to be published hopefully in the same issue, if not in the following one. They will act as a secondary source, and then we will restore the content and you will have no excuses. If you are an applied mathematician, this means you are either never work with ML algorithms, or you did but you suffer some bad feelings. :) 85.143.254.228 (talk) 05:42, 6 November 2023 (UTC)[reply]

Multiple editors removed the content, and were reverted by various anonymous IP addresses, which I strongly suspect were the same person. In any case, once the new material was reverted, the burden was on you to build consensus before restoring it. As I've already said, our policies advise against this content without much better sources. But if, as you suggest, this identity is of central importance to ML, then there should be some very good secondary sources on the topic (e.g., Machine learning textbooks). I would consider an article by Hesham Sharkas as a primary source, and in any case our policy specifies that typically independent secondary sources are required. Sharing an author makes them not independent. Tito Omburo (talk) 10:21, 6 November 2023 (UTC)[reply]

Mistake in "recovering the Fourier Series form of Parseval's Identity"

Hello,

The proposed basis $e_{n}=e^{-inx}$ for recovering the Fourier series form of Parseval's Identity is incorrect. Indeed, the proposed basis is not orthonormal in $L^{2}[-\pi ,\pi ]$ as $\langle e_{n},e_{n}\rangle =2\pi$ . Moreover, the inner product in $L^{2}[-\pi ,\pi ]$ conjugates the second argument. Therefore, in order to recover the formula for the Fourier coefficients in the article, I believe it should be $e^{inx}/{\sqrt {2\pi }}$ .

Hope this helps. XMissingno (talk) 11:55, 12 January 2023 (UTC)[reply]

Merge with Parseval's theorem

It appalls me that these are separate pages in the first place - sure, Parseval's theorem and Parseval's identity are conceptually distinct, but once is essentially a corollary of the other. Is there a particular reason why these are separate, or is it just an oversight -- in which case a merge would be adequate?

96.35.171.223 (talk) 07:11, 3 February 2012 (UTC)[reply]

Agreed. I have so proposed in the most recent edit to both pages. Jim Bowery (talk) 20:53, 15 December 2023 (UTC)[reply]

For reference, the other article this discussion concerns is Parseval's theorem. Tito Omburo (talk) 16:57, 19 December 2023 (UTC)[reply]

Support. Based on the current article content, a merge is totally appropriate. I think Parseval's theorem should be the main article title, because I believe the scope might be made more general. Tito Omburo (talk) 16:59, 19 December 2023 (UTC)[reply]